# 2.16 Transpositions

## 2.16.1 Definition of a transposition

###### Definition 2.16.1.

A transposition is a 2-cycle.

For example, the only transpositions in $S_{3}$ are $(1,2)$, $(2,3)$, and $(1,3)$.

## 2.16.2 Every permutation is a product of transpositions

In this section we’re going to prove that every permutation can be written as a product of transpositions. Before we do so, here is some motivation for why we should expect this to be true.

One way we’ve already seen of illustrating a permutation $s$ in $S_{n}$ is to draw the numbers $1,2,\ldots,n$ in a column, and then in another column to its right, and draw a line from each number $i$ in the first column to $s(i)$ in the second. You get what looks like a lot of pieces of string tangled together.

Here is a diagram of the permutation $s=(1,2,3,4,5)$. Think of the lines as pieces of string connecting $i$ on the left with $s(i)$ on the right.

Composing two permutations drawn in this way corresponds to placing their diagrams side-by-side:

Imagine taking such a diagram and stretching it out. You could divide it up into smaller diagrams, each of which contains only one crossing of strings.

A diagram with a single string crossing is a transposition, since only two numbers change place. The diagram above illustrates the fact that

 $(1,2,3,4)=(1,2)(2,3)(3,4).$

Now we’re ready for a formal proof of the result that every permutation equals a product of transpositions. We first do it for cycles:

###### Lemma 2.16.1.

Every cycle equals a product of transpositions.

###### Proof.

Let $a=(a_{0},\ldots,a_{m-1})$ be a cycle. Lemma 2.14.1 tells us that

 $a=(a_{0},a_{1})(a_{1},a_{2},\ldots,a_{m-1}).$

Using that lemma again on the cycle $(a_{1},\ldots,a_{m-1})$ we get

 $a=(a_{0},a_{1})(a_{1},a_{2})(a_{2},a_{3},\ldots,a_{m-1}).$

Repeating this gives

 $a=(a_{0},a_{1})(a_{1},a_{2})\cdots(a_{m-2},a_{m-1})$

which shows that $a$ can be written as a product of transpositions. ∎

To illustrate this result you should check by computing both sides that

 $(1,2,3,4)=(1,2)(2,3)(3,4).$
###### Theorem 2.16.2.

Every permutation in $S_{n}$ is equal to a product of transpositions.

###### Proof.

Let $p$ be a permutation. We have seen that every permutation can be written as a product of cycles, so there are cycles $c_{1},\ldots,c_{k}$ such that $p=c_{1}\cdots c_{k}$. The lemma above shows how to write each $c_{i}$ as a product of transpositions, which expresses $p$ as a product of transpositions too. ∎

###### Example 2.16.1.

Suppose we want to express

 $s=\begin{pmatrix}1&2&3&4&5&6\\ 2&3&1&5&6&4\end{pmatrix}$

as a product of transpositions. A disjoint cycle decomposition for $s$ is

 $s=(1,2,3)(4,5,6)$

and applying the lemma above, we get

 $\displaystyle(1,2,3)$ $\displaystyle=(1,2)(2,3)$ $\displaystyle(4,5,6)$ $\displaystyle=(4,5)(5,6)$

So

 $s=(1,2,3)(4,5,6)=(1,2)(2,3)(4,5)(5,6).$